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Growth and Welfare Effects of Intellectual Property Rights when Consumers Differ in Income Kiedaisch, Christian DOI: https://doi.org/10.1007/s00199-020-01322-9 Publication date: 2020 Document Version
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Kiedaisch, C 2020 'Growth and Welfare Effects of Intellectual Property Rights when Consumers Differ in Income' Economic Theory. https://doi.org/10.1007/s00199-020-01322-9
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Growth and Welfare Eects of Intellectual Property Rights
when Consumers Dier in Income
*
Christian Kiedaisch
October 25, 2019
Abstract
This paper analyzes how changing the expected length of intellectual property (IP) protection T aects economic growth and the welfare of rich and poor consumers. The analysis is based on a product-variety model with non-homothetic preferences and endogenous markups in which, in accordance with empirical evidence, rich households consume a larger variety of goods than poorer ones. It is shown that growth is independent of T when there is perfect equality and that T can only substantially aect growth when there is a su cient degree of inequality. When there is inequality, an increase in T that is applied to both new and previously granted innovations increases growth. A reduction in T that aects only new, but not previously granted innovations, can increase growth if wealth inequality is su ciently high. In the case where increasing T increases growth, poorer households prefer a shorter length of protection T than richer ones. (JEL O34, O31, L16, D30, O15)
Keywords: intellectual property rights, income distribution, endogenous growth, non-homothetic preferences
1 Introduction
With income and wealth inequality on the rise in many developed countries (see for example Piketty, 2014), distributional concerns are taking center stage in many policy *I gratefully acknowledge nancial support by the Swiss National Science Foundation grant
CR-SII1_154446 "Inequality and Globalization: Demand versus Supply Forces and Local Outcomes". I thank Gilles Saint-Paul, Franck Portier, André Grimaud, Vincenzo Denicolò, Reto Föllmi, Josef Zweimüller, Josef Falkinger, Fabrizio Zilibotti, Kiminori Matsuyama, Iain Cockburn, Manuel Amador, Stefan Legge, Holger Strulik and Christian Catalini for helpful discussions and seminar/ conference par-ticipants in Toulouse, Ascona, Aix-en-Provence, Vigo, Barcelona, Heidelberg, Boston and Zürich for helpful comments.
Contact: CERPE-DeFiPP, Université de Namur, Rempart de la Vierge 8, 5000 Namur, Belgium,
debates. When it comes to changes in intellectual property right (IPR) policies like those included in the recently signed CPTPP trade agreement, it is therefore important to understand what their distributional consequences are and whether their eect on innovation and growth depends on the extent of inequality.
While strengthening intellectual property (IP) protection can have direct distributive eects by raising the relative wages of workers performing R&D intensive tasks, this paper studies interactions between inequality, IP protection and the demand for new goods. In standard product-variety growth models (e.g. Romer, 1990, or Grossman and Helpman, 1992), the assumption of homothetic preferences (that are identical across households) implies that the demand for individual goods and the incentives to innovate do not depend on the distribution of income. The implication of these models that rich households consume the same variety of goods as poorer ones and just proportionally more of each good, is, however, at odds with empirical evidence: Jackson (1984) and Falkinger and Zweimüller (1996) nd that the variety of goods consumed increases in household income. Based on data from the US consumer expenditure survey (CEX), Figure 1 shows that there is also a positive association between a more narrowly de ned variety of innovative goods purchased by a household and household expenditures.
Figure 1: 2 4 6 8 10 12
average number of innovative goods purchased
1 2 3 4 5 6 7 8 9 10
deciles of total expenditures per household
Innovative Goods Variety by Expenditure Group, 2012
Notes: Out of the over 600 goods from the CEX (INTR), 61 were classi ed as innovative, among them goods like computer software, video game hardware, portable memory, televisions, photographic equipment and new cars. A complete list of the selected goods is provided in Appendix B1. The number of innovative goods is de ned as the number of these selected goods of which a household has purchased at least one unit in 2012. Population weights are those representative for the US population. I thank Liliya Khabibulina for providing this graph.
As rich and poor households dier considerably with respect to their consumption pattern1, it is plausible that the demand for individual innovative goods depends on
the distribution of income. If rms that invent new goods rely on IPRs in order to protect their innovations, their pro ts and incentives to innovate should therefore depend on both the extent of IP protection and the degree of inequality across consumers. Moreover, it is likely that an increase in IP protection which leads to increased markups for some goods aects the consumption of rich and poor households in dierent ways.
This paper analyzes how varying the expected length of IP protection (also simply referred to as the length of protection) aects growth and the welfare of rich and poor con-sumers in a general equilibrium product-variety model with non-homothetic preferences that are identical across households. The analysis is based on Föllmi and Zweimüller (2006), who analyze the eects of inequality on growth, assuming in nite IP protection. Non-homotheticity is in this setting modeled in a simple and tractable way byassuming that goods are consumed in discrete amounts and that households are saturated after consuming one unit of a good. This assumption aligns with the observation that many of the existing goods are indivisible in nature and that usually only one unit of them needs to be consumed (think for example of fridges, TVs, mobile phones, laptops, cars, dierent types of software, etc.). Moreover, the assumption implies that, in line with the empirical evidence, rich households consume a larger variety of goods than poorer ones. The main dierence with respect to the previous literature is therefore that the analysis focuses on how IPRs in uence growth and welfare by aecting the extensive consumption margins (i.e. the varieties of goods consumed by dierent households) instead of the intensive consumption margins (which are absent due to the assumption of unit consumption).
In the model, dierences in labor productivities are exogenously given and there is free entry into R&D. Firms with IP protection engage in monopoly pricing, while goods are sold at marginal cost once IP protection has expired. IP protection is assumed to expire stochastically and, unlike in the previous literature, the general case is considered in which IP policy can have a dierential impact on the expiration rates of newly and previously granted IPRs. The model therefore allows to analyze realistic changes in IP policies that only aect IPRs granted after a certain date but not those granted before this date.
Analytical tractability is maintained by parametrizing the distribution of wealth in 1
In the CEX data, there is also a positive association between the variety of the remaining non-innovative goods and household expenditures, but no clearcut relation between household expenditures and the ratio between the variety of innovative and non-innovative goods that a household purchases. The basic model does not make any predictions about this ratio as it considers the case where all goods are innovative and only distinguishes between goods that are protected by IPRs and others that are not (a characteristic on which the CEX provides no information). In footnote 40 and in Section 5 extensions are discussed in which there are also non-innovative goods and in which rich households consume a larger variety of those goods than poor households.
a special way. This new modeling trick allows to derive all results taking transitional dynamics into account. It is important to account for those dynamics as changes in the strength of IP protection lead to changes in the distribution of wealth, so that their eects cannot be understood by simply comparing steady states, holding the extent of inequality constant. Föllmi and Zweimüller (2006) instead only consider the special case in which initial wealth is distributed in the same way as labor endowments, and in which an exogenous change in the distribution of labor endowments is always accompanied by a corresponding exogenous change in the distribution of wealth.
When there is perfect equality in the sense that all households have the same labor endowments and wealth levels, the rate of growth is independent of the level of IP protec-tion. The reason for this is that competition among monopolistic rms implies that they set their markups at such a level that each household can always aord to consume one unit of each existing good at each point in time. The extensive consumption margin is therefore xed. As there is no intensive consumption margin, this leaves a xed amount of spending that can be devoted to newly invented goods and implies that the total (present and future) revenues that can be earned by innovators are independent of the strength of IP protection. As a consequence, there is a markup eect according to which increases in the length of IP protection go along with corresponding reductions in markups and per period pro ts, leaving the incentives to innovate unchanged. Put dierently: as an increase in the length of IP protection implies that a larger fraction of the newly invented goods is (or will in the future be) sold at monopoly prices (instead of at marginal cost), these monopoly prices have to be correspondingly lower as the total revenues accruing to innovators remain unchanged. When unit consumption is a prevalent feature, this markup eect, which does not arise in standard models with homothetic preferences (in which markups are xed) can therefore substantially alter the eect that the length of IP protection has on growth .
When there are two groups of households with dierent incomes and expenditures, markups still adjust in such a way that rich households can consume one unit of each existing good in equilibrium. Poor households, however, just consume a fraction of these goods so that their extensive consumption margin is endogenous. In this setting with inequality, the eect of IP protection on growth can be broken down into two compo-nents: into its eect when expenditure dierences are given and its eect on expenditure dierences (which themselves aect growth).
When dierences in expenditures across households are given, an increase in the length of IP protection increases growth and goes along with a reduction in poor household's consumption. As the latter allows for an increase in the total revenues earned by innova-tors, the markup eect due to which markups fall in the length of IP protection is now less strong than in the case of perfect equality. An increase in the length of IP protection consequently increases the incentives to innovate and the rate of growth in this case. A
change in the length of IP protection canlead to a change in the dierences in expendi-tures across households and thereby in markups if it changes the value of initial wealth held in the form of non-expired IPRs. Such an eect is referred to as a valuation eects. If the length of IP protection is uniformly increased for both new and previously granted IPRs, the value of initial wealth remains unchanged (upon impact), implying that there is no such valuation eect and that increasing IP protection always increases growth as it does not aect expenditure dierences. .
An important feature of the model is that increases in the dierences in expenditures across households lead to an increase in markups and - as in Föllmi and Zweimüller (2006) - increase the rate of growth. Due to this mechanism, new feedback eects can emerge through the valuation eect when the length of IP protection is changed in a non-uniform way: if the length of IP protection is unexpectedly reduced for future innovations but not for previously issued IPRs, the value of the latter increases due to increasing markups. If the initial wealth consisting of previously issued IPRs is unequally distributed, this leads to an increase in the dierences in expenditures across households, which again implies a further increase of markups. In the case where the distribution of wealth is su ciently unequal, the latter eect can be so strong that 4.The presence of the valuation eect can consequently reverse the eect that IP protection has on growth. This is not the case in traditional models with (identical) homothetic preferences, in which markups are constant and in which neither the rate of growth nor the eects of IP protection on growth depend on the distribution of income. Given that consumption indivisibilities and satiation are considered to be relevant features, the model therefore suggests that it is very important to take valuation eects due to which changes in IP policies can change the distribution of wealth and the demand for new goods into account. In reality, households only hold a fraction of their wealth in the form of intangible assets the value of which to some extent depends on the strength of IP protection (or on corresponding antitrust policies). As the value of these assets is considerable (see Haskel and Westlake, 2017), the valuation eect might nevertheless be of non-negligible size.
Comparing the cases of perfect equality and inequality, the analysis shows that changes in the length of IP protection can only have substantial eects on the rate of growth if there is a su cient degree of inequality. This is the rst main result of the paper.
In the case where there is inequality and where an increase in the length of IP pro-tection increases growth, it reduces the variety of goods consumed by poorer households more (in absolute terms) than that consumed by richer households, even though the lat-ter consume a larger variety of IP protected goods and therefore contribute more to the monopoly incomes of innovators. The intuition for this surprising result is the following: while all households consume all of the goods on which IP protection has expired (as those are the cheapest ones), implying that their expenditures are eectedin a similar way when there are fewer of these goods, richer households bene t more from the decline
in markups of IP protected goods (i.e. from the markup eect), as they consume a larger variety of these goods. As poorer households bene t similarly to richer ones when the rate of growth increases due to an increase in the length of IP protection but suer from a larger reduction in current consumption, they prefer a shorter duration of IP protection in this case than richer households do. This is the second main result of the paper. Even if IP policies do not aect inequality through supply side forces or the valuation channel, they can therefore lead to con icts of interest when rich and poor households dier with respect to their consumption pattern. Agreeing on a uniform (international) length of IP protection might therefore be a di cult task, as poor households or countries might need to be compensated appropriately in order to support long IP protection2. The
in-volved distributional con ict can be substantial: in the case of two income groups, poor households actually bear the whole costs in terms of reduced current consumption when the length of IP protection is increased while the consumption of rich households remains unchanged.
A further result of the paper is that an increase in the discount rate can lead to an increase in the rate of growth when wealth inequality is su ciently large. The reason for this is that a larger discount rate implies an increased propensity to consume out of interest income and that it can therefore increase the inequality in consumption expen-ditures. As the latter implies larger markups and pro ts, the value of an innovation can actually increase even though future pro ts are discounted more heavily. Such a paradox of thrift does not arise in standard growth models.
In an extension where rms´ price setting power is reduced since households can also consume substitutable non-innovative goods, it is shown that inequality no longer needs to encourage growth and that it is more likely to discourage growth, the weaker IP protection is. When markups can be reduced through a reduction in the breadth of IP protection, it is shown that, holding the rate of growth constant, rich households prefer long and narrow IP protection, while poor households prefer short and broad protection. The paper is structured as follows: In Section 2, the related literature is discussed. Section 3 describes the model setup and Section 4 analyzes the equilibrium. In Section 4.3, the case of perfect equality is analyzed, while sections 4.4 - 4.7 focus on the case of two income groups. Section 4.8 studies the welfare eects of IPRs. Section 4.9 extends the analysis to the case where there are many income groups and applies it to an international context. In Section 5, the extension of a limited price setting power is discussed. Section 6 concludes. Proofs are collected in Appendix A and extensions and supplementary material in Appendix B.
2In the basic setup of the model, there is no e cient length of IP protection on which all households
2 Related literature
As this paper builds on Föllmi and Zweimüller (2006), the main dierences have already been addressed in the introduction. Two more dierences should be mentioned: while Föllmi and Zweimüller (2006) study the case of hierarchical intra-temporal preferences and assume that intertemporal utility is of the CES type, the present paper focuses on the special case in which goods are symmetric and in which intertemporal preferences are logarithmic. This is done in order to keep the analysis simple and tractable. An extension with hierarchical preferences in which innovation is endogenously targeted towards rela-tively more luxurious goods and in which non-IP protected goods serve relarela-tively more basic needs is analyzed in Appendix B5.
Building on and correcting the analysis of Kwan and Lai (2003), Cysne and Turchick (2012) study the optimal stochastic IP expiration rate in a lab-equipment product vari-ety model based on Rivera-Batiz and Romer (1991). Taking transitional dynamics into account, they nd that growth increases when the (expected) length of IP protection is increased3. Unlike the present paper, they, however, do not consider the case where the
IP expiration rate can dier for newly and previously granted IPRs. Cysne and Turchick (2012) nd that full IP protection is optimal if preferences over nal goods are logarith-mic, but not necessarily if preferences are of a more general CES type. In this setup, markups are determined by the constant and exogenously given elasticity of substitu-tion between goods and, unlike in the present paper, do not depend on the length of IP protection or the distribution of income4.
Chu (2010) studies a quality ladder model with homothetic preferences in which wealth is unequally and labor incomes are equally distributed. He nds that a (uniform) increase in patent breadth (modeled as an increase in markups) increases the rate of growth, g, and the rate of interest, r, and that the latter leads to an increase in income inequality. Furthermore, he shows that an increase in patent breadth increases (decreases) consump-tion inequality if the elasticity of intertemporal substituconsump-tion, ε, is smaller (larger) than one. The reason for this is that the increases in g and r lead to an increase (decrease) in the propensity to consume out of wealth, r − g, if ε < 1 (> 1). The present paper instead focuses on the case where ε = 1 (logarithmic intertemporal preferences), in which changes in r and g do not change the inequality in household expenditures through this channel. Using the setting of Chu (2010) but assuming that R&D uses nal goods instead 3In standard product variety models, increasing the (expected) length of IP protection always increases
steady state growth (see for example Barro and Sala-i-Martin, 2004, Chapter 6.2). Exceptions are Furukawa (2007) and Michel and Nyssen (1998) in which growth can be maximal under nite patent protection as the former assumes that patent protection reduces learning by doing in the nal goods sector and the latter that it prevents other R&D rms from accessing the patented knowledge.
4Föllmi and Zweimüller (2002) introduce hierarchical preferences into a product-variety growth model
with a representative consumer. They nd that markups for patent protected goods rise over time and that this implies that utility along a balanced growth path is maximal for a nite patent length.
of labor as inputs, Chu and Cozzi (2018) show that a uniform increase in patent breadth cannot only increase inequality by increasing the interest rate, but also by lowering real wages while at the same time leaving the value of initial wealth unchanged. In the present paper, R&D is undertaken by labor5, so that this mechanism is not at work.
While the level of inequality also depends on the extent of IP protection in Chu (2010) and Chu and Cozzi (2018), it - unlike in the present paper - does not aect the incentives to innovate in their settings due to the assumption of homothetic preferences. This assumption implies that (unlike in the present model) households with dierent income levels but equal ratios of initial wealth to labor income face the same trade-o between growth and current consumption and would therefore prefer the same level of IP protection6.
A mechanism left out of the current analysis is that the level of IP protection might directly aect the distribution of labor incomes. Saint-Paul (2004) studies this mechanism in a two-period model in which only workers with a su ciently high level of skills decide to become R&D workers and in which an increase in IP protection increases the skill premium7. While a reduction in IP protection might bene t poor unskilled workers
(by reducing prices) and harm rich skilled workers (by reducing their wages), Saint-Paul (2004) argues that a social planner would always prefer to fully enforce IPRs and to address distributional concerns via redistributive taxation. While Saint-Paul (2004) derives the main results assuming homothetic preferences, he also shows the following: if a reduction in IP protection leads to a marginal reduction in the variety of goods that are invented and also in the fraction of monopolistically supplied goods, then poor households are more likely to bene t from it than richer ones if the elasticity of utility with respect to consumption increases in the level of consumption8. This result is, however, derived
under the assumption that incomes and markups are exogenously given.
In Saint-Paul (2006) the utility derived from the consumption of any single good is bounded from above, implying that, unlike in the case of CES preferences, rich house-5This is a natural speci cation in the case of non-homothetic preferences in which there is no
repre-sentative nal good (bundle) that could be used as an R&D input.
6This is, however, not shown by the authors. In the setup of Chu (2010), it can, moreover, be shown
that in the case where ε = 1, all households would prefer the same patent breadth, independent of their wealth to labor income ratio.
7Spinesi (2011) and Bernal Uribe (2012) derive similar results in growth models. In Cozzi and Galli
(2014) changes in IP legislation that increase the blocking power of basic research relative to applied follow-on research can either increase or decrease growth and the skill premium. In a model with directed technological change, Pan, Zou and Li (2015) nd that it is optimal to grant broader patent breadth to skill-complementary innovations and that this encourages skill-biased technological change and leads to an increase in wage inequality.
In empirical analyses based on cross country panel data, Adams (2008) and Saini and Mehra (2018) nd that strengthening intellectual property protection tends to be associated with increased income inequality. Using US cross-state panel and cross commuting-zone data, Aghion et al. (forthcoming) nd a positive eect of innovation on top income inequality.
8That means if cu0(c)
u(c) increases in c, where u(c) denotes the utility derived from a single good (total
holds value an increase in product variety (through innovation) more than poorer ones. While Saint-Paul (2006) does not study the role of intellectual property rights, he shows (in a working paper version, 2002) that a social planner who puts equal weight on each consumer prefers to allocate fewer resources to R&D than are actually allocated in equi-librium if patents are in nitely lived.
As all of the above-mentioned papers except for Föllmi and Zweimüller (2006) only analyze cases in which each household consumes positive quantities of each good, they, unlike the present paper, are at odds with the empirical evidence indicating that the variety of goods purchased increases in household income. They, moreover, do not take into account that the distributional eects of IP policies might to a large extent be driven by their eects on the extensive consumption margins (i.e. on the varieties of goods consumed by dierent households).
In Kiedaisch (2009b), I study a two-period product-variety model with strictly hi-erarchical preferences and sector-speci c cost-saving innovations. While an increase in IP protection always increases innovation when pro t and labor incomes are equally dis-tributed, there are two opposing eects when all pro t income accrues to a small minority of rich households: then, stronger IP protection encourages innovation in basic need sec-tors but - by reducing the purchasing power of mass consumers - reduces demand and the incentives to innovate in sectors that produce more luxurious goods, implying that overall innovation might be reduced. Unlike in the present paper, markups are con-stant in Kiedaisch (2009) and, due to the lack of free entry into R&D, an increase in IP protection increases the rents of rm owners and therefore has a more direct eect on inequality.
Within an international context, several papers have analyzed the eects of having dierent levels of IP protection in dierent countries that interact in a global economy9.
Contrary to this literature, the present paper focuses on the question whether dierences in the structure of demand can lead to disagreement about the optimal uniform global strength of IP protection.
Furthermore, the paper relates to an extensive literature about the relationship be-tween inequality and growth and speci cally to a few papers in which inequality aects innovation and growth through the channel of demand10.
9Chung and Lu (2014) analyze the eects of southern IP protection in a two-period North-South
model with hierarchical preferences in which innovators can supply goods of dierent quality levels. There is no within-country inequality in this model and, due to the lack of arbitrage, goods are sold at dierent prices in the North and the South. Kohler (2012) studies a dynamic North-South trade model with non-homothetic preferences and full IP protection. He nds that preventing arbitrage across countries reduces innovation and harms the rich North while it might bene t the poor South. Chu and Peng (2011) extend the model of Chu (2010) to a two-country context. Further references can be found in a recent literature survey by Saggi (2016).
10See Murphy et al. (1989), Falkinger (1994), Zweimüller (2000), Mastuyama (2002), Chou and
Tal-main (1996), Föllmi and Zweimüller (2006), Föllmi and Zweimüller (2016), Würgler (2010) and Föllmi, Würgler and Zweimüller (2014) for theoretical contributions. In an empirical study, Beerli et al
(forth-3 The model setup
3.1 Preferences and technology
There is a continuum of potentially producible dierentiated goods indexed by j ∈ [0, ∞). In a given period t, only one or zero units of any of these goods can be consumed by a household i: ci(j, t) ∈ {0, 1}.
Households are in nitely lived and intertemporal utility is given by: Ui(τ ) = ∞ ˆ t=τ ln ∞ ˆ j=0 ci(j, t) dj e −ρ(t−τ ) dt (1)
where ρ > 0 denotes the rate of time preference. While preferences are homothetic in the intertemporal dimension, the strong assumption of indivisibilities in the consumption of goods ("0−1 consumption") is made in order to introduce non-homothetic intra-temporal preferences in a simple and tractable way.
The factors of production are homogeneous labor and the measure N(t) of goods that have been invented until point in time t (the stock of knowledge). Producing one unit of an invented good requires b
N (t) ≥ 0units of labor as an input and attaining an innovation
in a sector j is associated with xed R&D costs equal to F
N (t) > 0 units of labor. The
assumption that labor productivity in both the production and the R&D sector increases in this multiplicative way in the stock of knowledge N(t) is made in order to allow for continuous exponential growth11.
3.2 Intellectual property protection and prices
The labor market is assumed to be competitive and there is free entry into R&D. An innovator who has invented a good j obtains intellectual property (IP) protection on it, which allows him to exclude others from producing this good. The intellectual property right, however, does not allow appropriating any of the spillovers, which increase the productivity of both rms that produce other goods and of future innovators, implying
coming) nd that changes in the Chinese income distribution led to considerable changes in market sizes for dierent durable goods and that productivity increased in sectors in which demand increased. Study-ing US product-level data (2004 - 2015), Jaravel (2019) nds that growth in demand from high-income consumers implied a faster increase in product variety for rich than for poor households and that this went along with lower in ation rates for richer households.
11Unlike in standard growth models, the productivity of the production sector needs to increase in
N (t)as the assumption of consumption indivisibilities precludes the possibility to consume and produce less of each good when the number of goods increases. Only in the special case of a digital economy where b = 0, these spillovers are not required (see footnote 32). If there were no spillovers in the R&D sector, growth would be linear but the qualitative results would be the same. A simple two-period version of the model without any spillovers is studied in Appendix B8.
that there is a research exemption12. IP protection is assumed to expire with hazard
rate γ (that means with probability γdt in time interval dt), implying that the expected length of protection is equal to T ≡ 1
γ. IPRs are therefore in nitely lived (T = ∞) if
γ = 0 and not protected at all (T = 0) if γ → ∞13. The analysis allows for general adjustments in IP protection at a point in time t = t0: IPRs granted after t0 can expire
with hazard rate γ1 (instead of rate γ like before t0), while IPRs granted before t0 that
have not expired by t0 can expire with hazard rate γ0 from t0 onward. The constellation
γ1 6= γ0 = γ therefore captures the standard case in which changes in IP policies only
aect newly granted, but not previously granted IPRs, while the constellation γ0 6= γ1 = γ
re ects the opposite case where IP policies are only changed for previously granted IPRs (expropriation when γ0 > γ and retroactive extension when γ0 < γ). The case where
γ1 = γ0 6= γ is referred to as a uniform change in IP protection as it aects newly and
previously granted IPRs in the same way.
The assumption that IP protection expires stochastically and not after a xed time period is mainly made in order to keep the analysis of transitional dynamics tractable14.
It can, however, also be interpreted as a way to parametrize the expected time for which imperfect IPRs protect their owners from imitation when the latter can even occur before the legal end of IP protection is reached. Appendix B4 analyzes the case in which IP protection has a nite and deterministic duration and the case in which IPRs are in nitely lived but only enforced with a certain probability at the time of invention. When the analysis is restricted to a comparison of balanced growth paths, the qualitative results are in these cases the same as those obtained when IP protection expires at a stochastic rate.
After the IPR on a good j has expired, anyone can freely produce this good and it is supplied at marginal cost due to perfect competition 15. The market clearing wage is
denoted by w(t). In order to obtain constant prices for the competitively supplied goods, the wage of a productivity-adjusted unit of labor is normalized to one, implying that the wage for one unit of labor is normalized to w(t) = N(t). Due to this normalization, the marginal production costs of a good and therefore the price of goods on which IP protection has expired is given by p(j, t) = b. The xed R&D costs are also constant over 12As R&D productivity increases in the stock of knowledge N(t), future innovators bene t from the
R&D undertaken by previous innovators. IP protection could therefore be broadened by granting in-novators some blocking power over future inventions which would enable them to extract licensing fees from future innovators. In Appendix B3 it is shown that granting such extended IP protection would reduce the rate of growth along a balanced growth path.
13In the following, T will often simply be referred to as the length of IP protection.
14If patents are of nite duration, it becomes very di cult to solve for the transitional dynamics within
such models if time is continuous (see Judd (1985) and Cai and Nitta (2012); Deneckere and Judd (1992) and Matsuyama (1999) study models in which time is discrete and in which patents last for one period).
15The analysis focuses on intellectual property rights as the only factor granting a monopoly position.
If such a position can be obtained through other factors like trade secrecy or weak antitrust policies, the same analysis applies to these factors as long as there are the same spillovers and as long as the monopoly position can be lost due to imitation or stricter antitrust policies with hazard rate γ.
time and given by F , as R&D labor is competitively supplied. It is assumed that a rm with IP protection in sector j cannot observe its customers´ income and therefore cannot price discriminate between households with dierent willingness to pay.
Households are not nancially constrained and can borrow and lend at the interest rate r(t).
3.3 Distribution
The size of the population and the total labor endowment of the economy are normalized to 1. While all households have the same utility function, it is assumed that there are poor (P ) and rich (R) households with population shares β and 1 − β (with 0 ≤ β < 1. The case of many income groups is analyzed in Section 4.8). A poor household´s labor endowment is given by lP = ϑ(0 < ϑ ≤ 1) and that of a rich household by lR = 1−βϑ1−β ≥ 1,
as βlP + (1 − β)lR = 1 must hold. For a given ϑ < 1, lR therefore increases in β. A
household´s labor endowment pins down its labor income as a share of the average labor income in the economy. Starting from a given distribution, a distribution with a larger β or a lower ϑ can be obtained through a series of regressive transfers. Therefore, inequality in labor incomes is said to increase if β increases or if ϑ decreases16.
At the initial date t = τ, the economy is endowed with wealth in the form of previously granted non-expired IPRs, the value of which is equal to the expected discounted pro t income accruing to their owners. A rich household´s initial wealth is denoted by VR(τ )
and that of a poor household by VP (τ ), and it is assumed that VR(t) ≥ VP(t) ≥ 0holds17.
3.4 Consumption choices
The intertemporal budget constraint of a household of type i ∈ {R, P } is given by:
∞ ˆ t=τ N (t)lie−R(t,τ )dt + Vi(τ ) ≥ ∞ ˆ t=τ ∞ ˆ j=0 p(j, t)ci(j, t) dj e −R(t,τ ) dt (2) where R(t, τ) = t ˆ s=τ
r(s)dsis the cumulative discount rate between dates τ and t. The left hand side represents the discounted sum of wage income (note that w(t) = N(t)) plus the value of initial wealth; the right hand side denotes the discounted sum of consumption expenditures (p(j, t) → ∞ when a good is not yet invented). A household maximizes intertemporal utility (equation 1) subject to this budget constraint. As preferences are
16The Gini coe cient of labor incomes is given by G = β (1 − ϑ).
17The analysis can be extended to cases where VR(t) < VP(t), where Vi(t) < 0 for one of the groups
(debt), or where ϑ > 1. As long as the distribution of labor endowments and initial wealth is such that a rich household is overall richer than a poor household and spends more in every period, this does not change the (qualitative) results.
additively separable across periods, this maximization problem can be solved by applying two-stage budgeting: the household rst maximizes instantaneous utility by maximizing the measure of goods Ci(t) =
∞
ˆ
j=0
ci(j, t) djthat can be purchased with given expenditures
Ei(t)= ∞
ˆ
j=0
p(j, t)ci(j, t) dj in a period t18and then optimally allocates expenditures across
periods. The rst problem is solved by purchasing one unit of all goods the prices of which lie below the household's willingness to pay zi(t) and a non-negative measure of goods
with prices equal to zi(t)19. Then, a Lagrangian with the variables Ci(Ei(t)) and Ei(t)
can be set up and maximized with respect to Ei(t) in order to derive zi(t). The resulting
optimal consumption rule is given by:
ci(j, t) = 1 if p(j, t) < eR(t,τ )−ρ(t−τ )µ iCi(t) ≡ zi(t) 1 or 0 if p(j, t) = zi(t) 0 if p(j, t) > zi(t) (3) where µi is the Lagrange multiplier and represents the marginal utility of income at
the initial date τ. Given that rich households spend more on consumption in a given period than poor households, they also consume a larger variety (measure) of goods, i.e. CR(t) > CP(t). In equilibrium, the intertemporal budget constraints are satis ed with
equality (implying that zi(t) < ∞) and the willingness to pay of a rich household exceeds
that of a poor household, so that zR(t) > zP(t) (and µR < µP)20.
3.5 Monopoly pricing
A rm that has IP protection on good j sets the price p(j, t) in order to maximize pro ts. As zR(t) > zP(t), market demand for any good j in period t is given by a
step function (see Figure 2): for a price higher than the willingness to pay of a rich household (p(j, t) > zR(t)), there is no demand for the good; for a price equal to or below
the willingness to pay of a rich but above that of a poor household (p(j, t) ∈ (zP(t), zR(t)]),
demand is given by the population size of the rich, 1 − β; and for a price below or equal to the willingness to pay of a poor household (p(j, t) ≤ zP(t)) demand is equal to one
(the size of the whole population).
18As all goods enter symmetrically into the utility function, households only care about the total
measure of goods consumed.
19There is always a positive measure of such goods in equilibrium, implying that ∂Ci(t)
∂Ei(t) =
1
zi(t) holds. 20In an extension that is discussed in Section 5, zR(t) = zP(t)can hold in a particular regime (C3).
Figure 2: Market demand of a monopolistic rm zR# zp# 1&β# β# A# B#
In order to maximize pro ts, an IP holding rm then either sets p(j, t) = zR(t) and
sells only to rich households (point A in Figure 2) or charges p(j, t) = zP(t) < zR(t)
and sells to both rich and poor households (point B). In the rst case, pro ts are given by πR(t) = (1 − β) (zR(t) − b), and by πP(t) = zP(t) − b in the second case (note that
marginal production costs are equal to b). The rm therefore charges low prices and sells to the whole population if πP(t) > πR(t), charges high prices and sells only to rich
households if πR(t) > πP(t), and is indierent between the two strategies if πP(t) = πR(t).
3.6 Price structure and regimes
The subset of sectors in which IP protection has expired is denoted by M(t) < N(t) and the de nition m(t) ≡ M (t)
N (t) is used. While prices are equal to b in these sectors, the prices
in the sectors in which IPRs are protected (of which there is a measure N(t) − M(t)) depend on the distribution of income and other parameters of the model.
Given that rich and poor households spend the amounts ER(t) and EP(t) (< ER(t))
in period t, either CP(t) < CR(t) < N (t) or CP(t) < CR(t) = N (t) holds21. The case
where CP(t) < CR(t) < N (t) can arise if even rich households do not spend enough to
be able to purchase all invented goods, even if they are sold at marginal cost (e.g. if ER(t) < bN (t)). As households value all goods equally, competition between rms then
implies that no good is sold at a price that exceeds the marginal cost of b in equilibrium22.
21The case where CP(t) = CR(t) = N (t) cannot be an equilibrium for the following reason: If rich
households purchased the same measure of goods as poor households (at the same prices), they would not exhaust their budgets, implying that their willingness to pay for an additional good, zR(t), would be
in nitely large. Then, some rms would have an incentive to increase their price and to sell exclusively to rich households, implying that poor households would not purchase one of each of the invented goods anymore.
22It is not possible that rich households split into groups that consume dierent subsets of goods at
equal but positive markups as rms would then have incentives to cut prices a bit in order to attract demand from households who do not yet purchase their goods.
As even IP holding rms do not earn any pro ts in this case, there are no incentives to undertake costly R&D and there cannot be positive growth. The more interesting case in which IP holding rms earn positive pro ts arises when ER(t) > bN (t). Then,
CP(t) < CR(t) = N (t) holds, meaning that rich households consume one of each of the
invented goods while poor households just consume the fraction cP(t) ≡ CN (t)P(t) of those
goods (referred to as their consumption share). The following analysis focuses on this case in which two dierent regimes can emerge:
In Regime A, CP(t) > M (t)(Condition A) holds, so that poor households not only
consume goods the IPRs of which have expired but also some more expensive IP protected goods. Part of the IP protected goods are then exclusively sold to rich households at the price pR(t) = zR(t) while others are sold to both rich and poor households at the price
pP(t) = zP(t)(> b). As IP holding rms supply symmetric goods, they must be indierent
between both strategies so that πR(t) = (1 − β)(pR(t) − b) = πP(t) = (pP(t) − b) has to
hold. This implies that
pP(t) = zP(t) = βb + (1 − β)pR(t) (4)
Firms that sell to both groups therefore charge a price that is lower than the price pR(t) = zR(t) charged by rms that sell exclusively to rich households.
In Regime B, CP(t) < M (t) holds (implying that Condition A is violated), so that
poor households only consume goods the IPRs of which have expired. This regime arises if EP(t) < bM (t), i.e. if the spending of a poor household is small relative to the measure
M (t) of competitively supplied goods.
In the following sections the equilibrium of the dynamic model is derived and the endogenous variables N(t), M(t), Ei(t), Ci(t), pj(t), πj(t), r(t) and zi(t) are derived as
functions of the exogenous parameters.
4 The general equilibrium
This section studies the general equilibrium of the model. The analysis rst studies the special case of perfect equality (i.e. the limit case where β → 0) and then focuses on Regime A in which even poor households purchase some IP protected goods in equilib-rium, so that CP(t) > M (t)(Condition A) holds. Regime B, in which only rich households
4.1 The allocation of resources across sectors
In Regimes A and B, the demand for production labor LD in period t is given by
LD(t) = ∞ ˆ j=0 b N (t) [βcP(j, t) + (1 − β)cR(j, t)] dj = bβcP(t) + b(1 − β) as N (t)b units of
labor are needed in order to produce one unit of a good and as the population size of poor households is given by β and that of rich households by 1 − β. The simpli ca-tion arises as CR(t) =
∞
ˆ
j=0
cR(j, t) dj = N (t) and as poor households only consume a
subset cP(t) ≡ CN (t)P(t) of the existing goods. The demand LR for R&D workers depends
on how much research is undertaken, meaning on N (t) =• ∂N (t)∂t . As the invention of a new product requires F/N(t) units of labor, the demand for R&D workers is given by: LR(t) = F
•
N (t)
N (t) = F g(t), where g(t) denotes the rate of growth of the stock of knowledge
N (t). Equating supply and demand of labor in a given period yields 1 = LD(t) + LR(t).
Plugging the corresponding values into this equation and solving for g(t) gives the econ-omy's resource constraint:
g(t) = 1
F [1 − bβcP(t) − b(1 − β)] (5)
Given that b > 0, there is a negative relation between the rate of growth g(t) and the consumption share of poor households, cp(t). The reason for this is that, as rich
households always consume one of each of the invented goods in equilibrium, employing more workers in the R&D sector is only possible if fewer workers are used to produce goods for poor households.23
4.2 Value of an innovation and interest rate along a balanced
growth path
4.2.1
The expected value of an innovation Z(t) is equal to the expected discounted sum of pro t income that accrues to an IPR holder. Therefore, Z(t) =
∞ ˆ s=t π(s)e−R(s, t)ds, with˜ ˜ R(s, t) = s ˆ q=t
(r(q) + γ)dq denoting the cumulative discount rate between dates t and s that depends on both the interest rates and the hazard rate γ at which pro ts are lost due to expiring IP protection. Deriving zi(t) = pi(t) = e
R(t,τ )−ρ(t−τ )
Ci(t)µi with respect to time
23This negative relation between g(t) and cp(t)only disappears in the case of a digital economy where
gives the Euler equation: C˙i(t) Ci(t) = r(t) − ρ − ˙ pi(t) pi(t). As ˙ pR(t) pR(t) = ˙ pP(t) pP(t) according to equation 4, ˙ CP(t) CP(t) = ˙ CR(t) CR(t) = ˙ N (t)
N (t) = g(t)has to hold in any equilibrium, implying that as cP(t) = CP(t)
N (t)
must be constant. Along a balanced growth path (BGP), not only g(t) and cP(t), but
also per period pro ts π(t) = (1 − β)(pR(t) − b) = (pP(t) − b)and the willingness' to pay
zi(t) = pi(t) are constant. The Euler equation then implies that:
r(t) = ρ + g(t) (6)
The rate of interest is therefore constant along a BGP and positively related to the rate of growth and to the rate of time preference. Along a BGP, the expected value of an innovation is consequently given by
Z(t) = π r + γ =
(1 − β)(pR− b)
ρ + g + γ
Due to free entry into R&D, the value of an innovation Z has to be equal to the (wage) costs of innovating, which are given by F . Therefore, the following free entry condition needs to hold along a BGP with positive growth:
Z = (1 − β)(pR− b)
ρ + g + γ = F (7)
4.3 The case of perfect equality
In the following, the extreme case in which all household have the same labor incomes and wealth levels is analyzed. This case coincides with the case in which β = 0, i.e. in which there are no poor households. According to equation 4, the prices charged by IP holding rms are all the same and given by p = pR = pP in this case. The resource
constraint (equation 5) and the free entry condition (equation 7) then imply that the rate of growth is given by g(t) = 1
F [1 − b] and that p = 1 + F (ρ + γ).
Proposition 1. Suppose that 0 ≤ b ≤ 1. The rate of growth is then given by g(t) =
1
F [1 − b] and there are no transitional dynamics. The rate of growth is therefore
indepen-dent of the expected length T of IP protection. When newly granted IPRs expire at rate γ, all rms with non-expired IPRs charge the price p = 1 + F (ρ + γ).
Proof. The proof is contained in the proofs of propositions 3 and 6, looking at the case where β → 0.
What is the intuition for this surprising result? Following the same logic as in section 3.6, competition among IP holding rms implies that they adjust their markups in such a way that each of the households ends up consuming one unit of each invented good at each point in time. As this pins down the demand for production labor and (due to
the resource constraint) leaves a xed amount of labor for research, the rate of growth is xed.
The mechanism can also be understood by looking at the pro tability of R&D: an increase in the expected length T of IP protection (i.e. a reduction in γ) on the one hand increases the value of an innovation by increasing the expected time span during which an innovator has monopoly power, but on the other hand leads to a reduction in the price p and therefore to a reduction in markups and in per period monopoly pro ts. In the case of perfect equality, this markup eect is so strong that the two eects exactly oset each other, leaving the incentives to innovate unchanged24. The reason
for this is the following: it is only possible to induce more rms to pay the xed costs of R&D if the total expected discounted pro ts accruing to innovators increase. Pro ts can increase if either revenues increase or if production costs fall. As each household always consumes one unit of each existing (previously invented) good in equilibrium, this leaves a xed amount of expenditures that can be allocated to newly introduced goods25,
pinning down the total revenues that innovators can earn. As, due to the assumption of unit consumption, innovators end up selling one unit of each invented good to each households at each point in time, the per period production costs faced by each innovator are xed as well. This implies that the total pro tability of R&D is independent of T. Consequently, competition among IP holding rms for given revenues implies that changes in T merely shift pro t income across sectors (by aecting the fraction m of competitively supplied goods on which IP protection has expired) and across time (by aecting prices p and per period pro ts), without aecting the value of an innovation. The incentives to innovate and the rate of growth are therefore independent of T26.
The assumption of unit consumption is clearly crucial for obtaining the result that the rate of growth does not depend on the strength of IP protection. Suppose that households could instead consume more than one unit of each good, and that they would consume more units of the cheap goods on which IP protection has expired than of the more expensive IP protected goods. Then, an increase of T that leads to a reduction of the share m of cheap (non IP protected) goods would induce households to consume less units of existing (previously invented) goods and increase the resources available for 24For a similar reason, g is independent of ρ: an increase in discounting implied by an increase in ρ is
exactly oset by an increase in markups, leaving the value of an innovation unchanged.
25It should be noted that the value V (τ) of initial wealth in the form of non-expired IPRs that is
held by each household does not aect the expenditures that households can devote to new innovations. The reason for this is that V (τ) merely re ects the monopoly pro ts that households have to pay to themselves when they purchase previously invented IP protected goods at positive markups. How much expenditures households can allocate to new innovations therefore only depends on the dierence between their labor income and the total production costs of the previously invented goods that they consume (this relation is re ected in equation 5).
26In Appendix B8 it is shown that the result that the strength of IP protection does not aect
innova-tion when there is perfect equality can also be obtained in a simple two-period model in which incomes, marginal production costs and R&D costs are exogenously given. Although there is no (explicit) resource constraint in this simple model, the mechanisms that are at work are the same as those described above.
R&D and the rate of growth. By how much an increase in IP protection would increase growth would then depend on how quickly households get satiated with individual goods if they consume more units of them. The faster satiation sets in after the rst unit of a good is consumed, the closer one then gets to the case of unit consumption in which growth is independent of T27. The model presented here is therefore most appropriate to
describe situations in which households only consume one unit of each good (one fridge, one car, one mobile phone, one software of a certain type...)28
4.4 Equilibrium price structure with two income groups
In the following, the BGP values of m(t) ≡ M (t)
N (t) and pR are derived for the case with two
income groups.
Multiplying the measure N(t) − M(t) of sectors in which IPRs are protected with the hazard rate γ with which IP protection expires, the absolute increase in the measure M (t) of sectors in which IP protection has expired is given by ˙M (t) = γ (N (t) − M (t)). Taking into account that g(t) = N (t)˙
N (t), we can derive ˙m(t) = γ (1 − m(t)) − m(t)g(t).
Along a BGP, ˙m = 0 needs to hold, so that
m = γ
g + γ (8)
Given that cp > m holds (Condition A; Regime A), the BGP consumption
expendi-tures of a poor household in period t are given by
∞
ˆ
j=0
p(j, t)cP (j, t) dj = N (t) [mb + pP (cp− m)]
and those of a rich household by
∞
ˆ
j=0
p(j, t)cR(j, t) dj = N (t) [mb + pP(cP − m) + pR(1 − cP)]
27The result that markups and per period pro ts fall in T (the markup eect) can also change when an
intensive consumption margin is introduced: So, markups are independent of T when goods are divisible and when preferences are of the standard CES type. Deneckere and Judd (1992) and Matsuyama (1999) study models with CES preferences in which the elasticity of substitution between goods is so high that a reduction in the fraction of competitively supplied goods m (that could be caused by an increase in T ) actually increases the demand for IP protected goods and (ceteris paribus) the per period pro ts of IP holding rms.
28The result that markups and per period pro ts fall in T (the markup eect) can also change when an
intensive consumption margin is introduced: So, markups are independent of T when goods are divisible and when preferences are of the standard CES type. Deneckere and Judd (1992) and Matsuyama (1999) study models with CES preferences in which the elasticity of substitution between goods is so high that a reduction in the fraction of competitively supplied goods m (that could be caused by an increase in T ) actually increases the demand for IP protected goods and (ceteris paribus) the per period pro ts of IP holding rms.
While both rich and poor households consume the fraction m of non-IP protected goods and the fraction cP − m of IP protected goods that are sold at the low price pP,
only rich households consume the fraction 1 − cP of IP protected goods that are sold at
the high price pR. A graphical representation of the equilibrium price and consumption
structure is given in Figure 3.
Figure 3: Price structure and consumption in Regime A
CP#=cPN# CR=N## M=mN# pR# pP# b# Measure#of#goods# price#
The per period labor income of a poor household is given by w(t)lP = N (t)ϑand that
of a rich household by w(t)lR = N (t)1−βϑ1−β . Using the relation N(t) = N(τ)eg(t−τ ) and
equation (4), the intertemporal budget constraint of a poor household in period t = τ is therefore given by:
N (τ )ϑ
r − g + VP(τ ) = N (τ )
r − g[mb + (bβ + (1 − β) pR) (cP − m)] (9) For a rich household, the intertemporal budget constraint is given by:
N (τ ) (1 − βϑ)
(r − g) (1 − β) + VR(τ ) = N (τ )
r − g[mb + (bβ + (1 − β) pR) (cP − m) + pR(1 − cp)] (10) Multiplying both sides of equations 9 and 10 by r − g = ρ (see equation 6), we obtain the consumption expenditures in period τ on the right hand sides. Expenditures of a poor household in any period t = τ are therefore equal to EP(τ ) = N (τ )ϑ + ρVP(τ ) and those
of a rich household are equal to ER(τ ) = N (τ )1−βϑ1−β + ρVR(τ ) > EP(τ ). Along the BGP,
households therefore spend all their labor income (w(t)li = N (t)li) in each period and the
fraction (r − g) Vi(t) = ρVi(t) of their interest income rVi(t), implying that expenditures
Ei(t) and individual wealth Vi(t) grow at rate g. This implies that VVR(t)
P(t) =
VR(τ )
VP(τ ) holds,
i.e. that the relative distribution of wealth stays constant over time along a BGP and that it re ects the initial distribution, independently of lR
lP , the distribution of labor
As there is a measure N(t) (1 − m) of sectors in which IP protection has not yet expired and as the value of an IP protected innovation is given by Z = F (see (7)), the total value of initial wealth along a BGP is given by
V (t) = N (t) (1 − m) Z = N (t) (1 − m) F (11)
When V (t) changes due to a change in m, also VR(t) and VP(t) change. As will be
proven in Section 4.6 which studies the transitional dynamics, these changes in Vi(t) are
(unless IP protection is changed in a non-uniform way) of equal absolute size29. Because
of that, the following special parametrization of the wealth distribution is used:
VR(t) = VP(t) + XN (t) (12)
where X ≥ 0 is a parameter that stays constant along a BGP (but might jump during a transition). Using equation 11 and the relation V (t) = βVP(t) + (1 − β) VR(t), we can
derive the individual BGP wealth levels as
VR(t) = (1 − m) F N (t) + βXN (t) (13)
VP(t) = (1 − m) F N (t) − (1 − β) XN (t) (14)
This parametrization ensures that VR(t), VP(t) and N(t) all grow at rate g along a
BGP when X is constant, implying a constant relative wealth distribution (i.e. that
VR(t)
VP(t) =
VR(τ )
VP(τ )does not change over time). The analysis focuses on the case where VP(t) > 0
(Condition B) holds. Dividing equation 13 by equation 14 and solving for X gives X = (1−m)F VR VP−1 β+(1−β)VR VP
. For any xed BGP value of m, any particular wealth distribution
VR(t)
VP(t) can therefore be generated by setting X equal to this value (which increases in
VR(t)
VP(t)
and decreases in m).
The modeling trick of parametrizing the distribution of wealth in this special way makes the following analysis tractable. The reason for this is that a change in m resulting from a change in an exogenous parameter turns out to leave X ≡ VR(t)−VP(t)
N (t) unchanged
(unless a non-uniform change in IP protection is considered), while it leads to a change in the relative wealth distribution VR(t)
VP(t).
Plugging equation 12 into equation 10, subtracting equation 9 from equation 10, and solving for pR gives:
pR= 1−ϑ 1−β + ρX
1 − cP
(15) 29The reason for this is that both rich and poor households consume the same absolute measure N(t)m
pRtherefore increases in cP, β, X and ρ and decreases in ϑ. The mechanism behind these
results is the following: the entire expenditure dierence ER(t)−EP(t) = N (t) (lR− lP)+
ρ (VR− VP) = N (t)
1−ϑ 1−β + ρX
is used to purchase the measure N(t) (1 − cP) of IP
protected goods that are each sold at price pR. For a given expenditure dierence, pR
must therefore increase in cP, as otherwise the excess expenditures of rich households
could not be absorbed by the purchase of fewer highly priced goods. A reduction in ϑ and an increase in β increase the expenditure dierence by increasing the dierence in labor income N(t) (lR− lP), and an increase in X or ρ increases it by increasing the
dierence in consumption out of interest income. For cP given, this increased expenditure
dierence can only be absorbed if pR increases.
4.5 Properties of the BGP
Plugging equation 15 into equation 7 and solving for g, the free entry condition can be written as: g = 1−ϑ 1−β + ρX 1 − cP − b ! 1 − β F − ρ − γ (16)
The growth rate g depends positively on cP, X and β and negatively on ϑ, γ and F .
The reason for this is the following: an increase in cP, X, and β and a decrease in ϑ
increase the value of an innovation by increasing pR30 and an increase in γ reduces the
value of an innovation by shortening the length of IP protection. An increase in F , on the other hand increases the costs of innovating. As the value of an innovation decreases if the interest rate r increases due to an increase in the rate of growth (remember that r = ρ+g (equation 6)), the free entry condition, which equates the value of an innovation to the costs of innovating implies a positive relation between g and the value of an innovation and a negative relation between g and F . The eect of ρ on g depends on the size of X: For small values of X, the free entry condition implies a negative relation between g and ρ which is mainly driven by the fact that an increase in ρ reduces the value of an innovation through a discounting eect. For large values of X, this discounting eect can, however, be dominated by a positive price eect that results from an increase in the expenditure dierence between rich and poor households: As an increase in ρ increases the consumption out of interest income ρVi(t), it also increases the expenditure dierence
and therefore pR and this eect is stronger the larger X is.
The free entry condition (equation 16) together with the resource constraint (equation 5) determine the general equilibrium.
30In the case where β increases, this eect is weakened by a reduction in the relevant market size,
1 − β. However, the value of an innovation still increases as the rst eect is stronger than the second one.
Proposition 2. A balanced growth path (BGP) in Regime A exists if 0 ≤ X ≤ Xe, 0 < b < 1−β1 , (1 − β) ρX − F (ρ + γ) < ϑ < (1 − β) ρX + F (ρ+γ)(1−b)bβ + 1−bβ + b, and T ≡ γ1 > eT hold, where ˜X and ˜T de ne positive and nite threshold values. Assuming that X ≡ VR(t)−VP(t)
N (t) remains constant when other parameters are changed, the following
holds:
a) Along the BGP, an increase in the expected length of IP protection T (≡ 1 γ) is
associated with an increase in the rate of growth g and with a reduction in the consumption share of poor households, cP.
b) Along the BGP, g depends negatively on ϑ and F and positively on β and X, so that inequality increases growth. cP depends positively on ϑ and negatively on X.
c) There is a positive threshold ˆX < ˜X such that for 0 ≤ X < ˆX, the BGP value of g depends negatively (and cP positively) on the discount rate ρ. For ˆX < X ≤ ˜X, an
increase in ρ is associated with an increase in the rate of growth g (paradox of thrift) and with a decrease in cP.
Proof. See Appendix A1
Contrary to the case of perfect equality, an increase in T (i.e. a fall in γ) increases the BGP growth rate when there are two income groups. How is this possible? While rich households still always consume one unit of each invented good (cR = 1), the extensive
consumption margin of poor households now depends on T . As a reduction of the con-sumption share cP of poor households reduces demand for production labor, more labor
can be allocated to the R&D sector, which leads to a larger rate of growth.
The mechanism can also be explained by looking at the pro tability of R&D: while the expenditures that rich households devote to newly invented goods remain unchanged, the reduction in cP implies that poor households dedicate a larger fraction of their
ex-penditures to newly invented goods. Therefore, total revenues accruing to innovators increase, allowing a larger number of them to break even. An increase in T reduces the fraction m of sectors in which IP protection has expired (see equation 8) and reduces the prices pP and pR at which IP protected goods are sold in equilibrium (see equations
15 and 4), so that there is again a markup eect. This eect is, however, less strong than in the case of perfect equality where total revenues accruing to innovators are xed. While both rich and poor households are equally aected by the declines of m and pP,
only the rich households bene t from the reduction in pRwhich allows them to purchase
the same measure of goods as before31.
31It should be noted that these results are not merely driven by the resource constraint (equation 5
which implies a negative relation between g and cP) and the fact that R&D uses labor as an input (and
not some nal composite good like in lab-equipment models) and also appear in a simple two-period version of the model in which incomes, marginal production costs and R&D costs are exogenously given (see Appendix B8).
An increase in inequality that results from a reduction in ϑ or from an increase in β increases the rate of growth. This result is a generalization of the result established by Föllmi and Zweimüller (2006) who (assuming that γ = 0) only consider the case in which initial wealth is distributed in the same way as labor endowments (VR
VP =
lR
lp =
1−βϑ (1−β)ϑ) and
in which an exogenous change in β or ϑ is accompanied by a corresponding exogenous change in VR
VP. If the wealth distribution becomes more unequal due to an increase in X,
this increases the rate of growth, as it increases the expenditure dierence between rich and poor households and allows innovators to raise prices32.
The paradox of thrift can arise for the following reason: an increase in the discount rate ρ increases the consumption out of interest income and, for X > 0, increases the expenditure dierence between rich and poor households, which allows innovators to raise prices. If wealth inequality (i.e. X) is large, this eect is stronger than the standard discounting eect due to which a rise in ρ discourages innovation, so that an increase in impatience can increase the rate of growth. Due to this mechanism, there might therefore be considerable growth in unequal economies with low savings propensities (like the US)33.
With the total labor income at date t given by N(t) and the total pro t income given by π (1 − m) N(t) = F (ρ + g + γ) g
g+γ
N (t) (equations 16 and 8 are used for the transformation), the labor income share can be derived as S = g+γ
g+γ+F g(ρ+g+γ). As ∂S
∂γ > 0 and ∂S
∂g < 0, an increase in the length of IP protection (i.e. a reduction in γ which
increases g) reduces the BGP value of S.
4.6 Transitional dynamics
While the previous section (Proposition 2) compared BGPs assuming X ≡ VR(t)−VP(t)
N (t) to
stay the constant, this section analyzes transitional dynamics, taking into account that X might change when other parameters are changed.
Suppose that the economy is on a BGP and that at date t = t0 there is an unexpected
change in one (or several) of the exogenous parameters. An (unexpected) change in the length of IP protection is parametrized as described in subsection 3.2: instead of expiring with hazard rate γ like before t0, IPRs granted after t0 expire with hazard rate γ1, while
IPRs granted before t0 that have not expired by t0 expire with hazard rate γ0 from t0
onward. Then, the following proposition holds:
32In the case where b = 0 (digital economy with zero marginal production costs), we obtain
g = F1 (see equation 5) and cP = 1 −1−ϑ+ρX(1−β)1−F (ρ+γ) along a BGP. While g is independent of inequality
in this case, cp still depends positively on γ and ϑ and negatively on X. Poor households therefore still
suer from an increase in the length of IP protection (while rich households are indierent) although it does not aect the rate of growth as all labor is inelastically supplied to the R&D sector.
33While this paradox can even occur in the case of full IP protection, it has not been derived by Föllmi
and Zweimüller (2006) who only analyze the case in which the distribution of wealth coincides with the distribution of labor income.
